Glossary of linear algebra
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This is a glossary of linear algebra.
See also: glossary of module theory.
A
- Affine transformation
- A composition of functions consisting of a linear transformation between vector spaces followed by a translation.[1] Equivalently, a function between vector spaces that preserves affine combinations.
- Affine combination
- A linear combination in which the sum of the coefficients is 1.
B
- Basis
- In a vector space, a linearly independent set of vectors spanning the whole vector space.[2]
- Basis vector
- An element of a given basis of a vector space.[2]
C
- Column vector
- A matrix with only one column.[3]
- Coordinate vector
- The tuple of the coordinates of a vector on a basis.
- Covector
- An element of the dual space of a vector space, (that is a linear form), identified to an element of the vector space through an inner product.
D
- Determinant
- The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of [math]\displaystyle{ 1 }[/math] for the unit matrix.
- Diagonal matrix
- A matrix in which only the entries on the main diagonal are non-zero.[4]
- Dimension
- The number of elements of any basis of a vector space.[2]
- Dual space
- The vector space of all linear forms on a given vector space.[5]
E
- Elementary matrix
- Square matrix that differs from the identity matrix by at most one entry
I
- Identity matrix
- A diagonal matrix all of the diagonal elements of which are equal to [math]\displaystyle{ 1 }[/math].[4]
- Inverse matrix
- Of a matrix [math]\displaystyle{ A }[/math], another matrix [math]\displaystyle{ B }[/math] such that [math]\displaystyle{ A }[/math] multiplied by [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B }[/math] multiplied by [math]\displaystyle{ A }[/math] both equal the identity matrix.[4]
- Isotropic vector
- In a vector space with a quadratic form, a non-zero vector for which the form is zero.
- Isotropic quadratic form
- A vector space with a quadratic form which has a null vector.
L
- Linear algebra
- The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.
- Linear combination
- A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element).[6]
- Linear dependence
- A linear dependence of a tuple of vectors [math]\displaystyle{ \vec v_1,\ldots,\vec v_n }[/math] is a nonzero tuple of scalar coefficients [math]\displaystyle{ c_1,\ldots,c_n }[/math] for which the linear combination [math]\displaystyle{ c_1\vec v_1+\cdots+c_n\vec v_n }[/math] equals [math]\displaystyle{ \vec0 }[/math].
- Linear equation
- A polynomial equation of degree one (such as [math]\displaystyle{ x = 2y - 7 }[/math]).[7]
- Linear form
- A linear map from a vector space to its field of scalars[8]
- Linear independence
- Property of being not linearly dependent.[9]
- Linear map
- A function between vector spaces which respects addition and scalar multiplication.
- Linear transformation
- A linear map whose domain and codomain are equal; it is generally supposed to be invertible.
M
- Matrix
- Rectangular arrangement of numbers or other mathematical objects.[4]
N
- Null vector
- 1. Another term for an isotropic vector.
- 2. Another term for a zero vector.
R
- Row vector
- A matrix with only one row.[4]
S
- Singular-value decomposition
- a factorization of an [math]\displaystyle{ m \times n }[/math] complex matrix M as [math]\displaystyle{ \mathbf{U\Sigma V^*} }[/math], where U is an [math]\displaystyle{ m \times m }[/math] complex unitary matrix, [math]\displaystyle{ \mathbf{\Sigma} }[/math] is an [math]\displaystyle{ m \times n }[/math] rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an [math]\displaystyle{ n \times n }[/math] complex unitary matrix.[10]
- Spectrum
- Set of the eigenvalues of a matrix.[11]
- Square matrix
- A matrix having the same number of rows as columns.[4]
U
- Unit vector
- a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.[12]
V
- Vector
- 1. A directed quantity, one with both magnitude and direction.
- 2. An element of a vector space.[13]
- Vector space
- A set, whose elements can be added together, and multiplied by elements of a field (this is scalar multiplication); the set must be an abelian group under addition, and the scalar multiplication must be a linear map.[14]
Z
- Zero vector
- The additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.[15]
Notes
- ↑ James & James 1992, p. 7.
- ↑ 2.0 2.1 2.2 James & James 1992, p. 27.
- ↑ James & James 1992, p. 66.
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 James & James 1992, p. 263.
- ↑ James & James 1992, pp. 80,135.
- ↑ James & James 1992, p. 251.
- ↑ James & James 1992, p. 252.
- ↑ Bourbaki 1989, p. 232.
- ↑ James & James 1992, p. 111.
- ↑ Williams 2014, p. 407.
- ↑ James & James 1992, p. 389.
- ↑ James & James 1992, p. 463.
- ↑ James & James 1992, p. 441.
- ↑ James & James 1992, p. 442.
- ↑ James & James 1992, p. 452.
References
- James, Robert C.; James, Glenn (1992). Mathematics Dictionary (5th ed.). Chapman and Hall. ISBN 978-0442007416.
- Bourbaki, Nicolas (1989). Algebra I. Springer. ISBN 978-3540193739.
- Williams, Gareth (2014). Linear algebra with applications (8th ed.). Jones & Bartlett Learning.
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